Basic Properties of Electric Charge

You already know that two types of charge exist and that opposite charges attract while like charges repel. But charge has deeper rules governing how it behaves. Three properties sit at the very foundation of everything you will study in electrostatics (and beyond): charges add up like simple numbers, charge can never be created or destroyed, and charge always comes in whole-number packets of a tiny fixed amount. Getting comfortable with these three ideas now will make every later topic, from Coulomb’s law to Gauss’s theorem, much easier to follow.

Before diving in, one quick idea: when the size of a charged body is extremely small compared to the distances separating it from other charges, we treat it as a point charge (a model where all the charge is imagined to sit at a single point in space). This simplification keeps the mathematics clean and is used throughout electrostatics.

Charges Add Up Like Ordinary Numbers: Additivity

Think of charge as a number on a number line. Positive charges sit to the right of zero, negative charges to the left. When you want the total charge of a system, you simply add all the individual charges together, respecting their signs.

If a system holds two point charges $q_1$ and $q_2$, the total charge is:

$Q = q_1 + q_2$

For a system with $n$ charges, this extends naturally:

$Q = q_1 + q_2 + q_3 + \ldots + q_n$

This property is called the additivity of charges. Charge is a scalar (a quantity with magnitude and sign, but no direction), so it adds up exactly the way ordinary real numbers do, just like mass. There is, however, one important difference between charge and mass: mass is always positive, whereas charge can be positive or negative. You must always include the correct sign when adding charges.

Here is a quick example to see this in action. Suppose a system contains five charges: $+1$, $+2$, $-3$, $+4$, and $-5$ (in some arbitrary unit). The total charge is:

$Q = (+1) + (+2) + (-3) + (+4) + (-5) = -1$

The positive and negative contributions partially cancel each other, and the leftover gives the net charge.

What Goes In Must Come Out: Conservation of Charge

When you rub a glass rod with silk, electrons move from the glass to the silk. The rod ends up positively charged and the silk negatively charged. Notice what did not happen: no new charge appeared out of thin air. Whatever the rod lost, the silk gained. The total charge of the rod-plus-silk system stayed exactly the same.

This observation holds universally and is one of the most firmly established laws in all of physics:

The total charge of an isolated system is always conserved. Charges within the system may shift around, redistribute, or transfer between objects, but the overall total never changes.

This is the law of conservation of charge, and it has been confirmed by countless experiments.

You might wonder: what about situations where charged particles themselves are created or destroyed? Nature does this all the time. For instance, a neutron (which is electrically neutral) can spontaneously decay into a proton ($+e$) and an electron ($-e$). Two charged particles have appeared where none existed before. Does this violate conservation?

Not at all. The proton’s $+e$ and the electron’s $-e$ add up to zero, exactly the charge the neutron had before the decay. The net charge before and after is the same. Charged particles may be born or destroyed, but the total charge of the system remains unchanged. It is simply not possible to create or destroy net charge.

Nature Counts in Steps: Quantisation of Charge

Here is a remarkable experimental fact: every free charge ever measured turns out to be a whole-number multiple of one tiny, fixed amount. That fixed amount is called the elementary charge, and it is denoted by $e$. In other words, the charge $q$ on any body always satisfies:

$q = ne$

where $n$ is an integer (positive, negative, or zero). You can have $1e$, $2e$, $-5e$, $137e$, but never $1.5e$ or $0.3e$. Charge does not come in arbitrary amounts; it comes in discrete packets.

This property is called the quantisation of charge.

Who discovered it?

The idea that charge is quantised was first suggested by the experimental laws of electrolysis (the process of using electric current to drive chemical reactions) discovered by the English experimentalist Faraday. Direct experimental proof came later, in 1912, from Millikan’s famous oil-drop experiment, which measured the charge on individual tiny oil droplets and showed that every measured value was an exact multiple of $e$.

The SI unit of charge and the value of $e$

In the International System of Units (SI), charge is measured in coulombs (symbol: C). One coulomb is defined as the amount of charge that flows through a wire in 1 second when the current is 1 ampere (A). Using this definition, the elementary charge works out to:

$e = 1.602192 \times 10^{-19} \text{ C}$

Flipping this around, a charge of $-1$ C would require roughly $6 \times 10^{18}$ electrons. That is a staggering number. In everyday electrostatics, you rarely encounter charges as large as a full coulomb, so smaller, more practical units are common:

• $1 \; \mu\text{C}$ (microcoulomb) $= 10^{-6}$ C
• $1 \; \text{mC}$ (millicoulomb) $= 10^{-3}$ C

The total charge on a real body

If a body contains $n_1$ electrons and $n_2$ protons, its total charge is:

$q = n_2 \times e + n_1 \times (-e) = (n_2 - n_1) \, e$

Since $n_1$ and $n_2$ are both integers, their difference is also an integer. So the charge on any body is automatically an integer multiple of $e$. Any change in the body’s charge happens in steps of $e$ as well: you can only add or remove whole electrons, never a fraction of one.

Why you do not notice quantisation in daily life

The step size $e = 1.6 \times 10^{-19}$ C is fantastically small. A charge of just $1 \; \mu\text{C}$ already contains around $10^{13}$ elementary charges. At this scale, adding or removing a single electron changes the charge by a negligible fraction. The tiny “graininess” of charge disappears completely, and charge appears to flow in a smooth, continuous stream.

A useful analogy: think of a dotted line drawn with very fine dots placed extremely close together. From a distance, it looks perfectly continuous. Only when you zoom in with a magnifying glass do you see the individual dots. Charge behaves the same way. At the macroscopic level (everyday objects, lab instruments), charge looks continuous. At the microscopic level (a handful of electrons or protons), the discrete, lumpy nature of charge becomes impossible to ignore.

So the bottom line is:

• Macroscopic scale (charges of $\mu\text{C}$ or larger): quantisation has no practical effect; charge can be treated as continuous.
• Microscopic scale (a few tens or hundreds of $e$): quantisation is essential; each charge is a countable, discrete packet.

The scale of the problem determines whether quantisation matters.

Worked Example 1.1: How Long to Collect One Coulomb?

Problem: If $10^9$ electrons leave a body every second and land on a second body, how long will it take for the second body to accumulate a total charge of 1 C?

Solution:

Step 1: Find the charge transferred each second.

Each electron carries a charge of $1.6 \times 10^{-19}$ C. With $10^9$ electrons moving per second:

$\text{Charge per second} = 10^9 \times 1.6 \times 10^{-19} \text{ C} = 1.6 \times 10^{-10} \text{ C/s}$

Step 2: Calculate the time to reach 1 C.

$t = \frac{1 \text{ C}}{1.6 \times 10^{-10} \text{ C/s}} = 6.25 \times 10^{9} \text{ s}$

Step 3: Convert seconds to years.

There are $365 \times 24 \times 3600 = 3.15 \times 10^7$ seconds in a year.

$t = \frac{6.25 \times 10^9}{3.15 \times 10^7} \approx 198 \text{ years}$

So even with a billion electrons leaving every single second, it would take roughly 200 years to build up just one coulomb of charge. This gives you a concrete feel for how enormous the coulomb really is compared to the tiny charges that appear in electrostatic experiments.

For perspective, a 1 cm cube of copper contains about $2.5 \times 10^{24}$ electrons. Even the huge number $10^9$ electrons per second is an incredibly tiny fraction of what sits inside an ordinary piece of metal.

Worked Example 1.2: Charge Inside a Cup of Water

Problem: How much total positive charge (and total negative charge) is contained in a cup of water?

Solution:

Step 1: Count the number of water molecules.

Assume the cup holds 250 g of water. The molecular mass of water ($H_2O$) is 18 g. One mole of water (18 g) contains $6.02 \times 10^{23}$ molecules (Avogadro’s number).

$\text{Number of molecules} = \frac{250}{18} \times 6.02 \times 10^{23}$

Step 2: Count the protons (and electrons) per molecule.

Each water molecule has 2 hydrogen atoms (1 proton each) and 1 oxygen atom (8 protons), giving 10 protons per molecule. Since the water is electrically neutral, it also has 10 electrons per molecule.

Step 3: Calculate the total positive charge.

$q_{+} = \frac{250}{18} \times 6.02 \times 10^{23} \times 10 \times 1.6 \times 10^{-19} \text{ C}$

$q_{+} \approx 1.34 \times 10^{7} \text{ C}$

The total negative charge has exactly the same magnitude: $q_{-} \approx 1.34 \times 10^{7}$ C.

That is a colossal amount of charge, more than 13 million coulombs, sitting inside a single cup of water. Yet you feel no electric force from it at all, because the positive and negative charges balance each other perfectly. This is a beautiful illustration of how charge conservation and electrical neutrality work together in ordinary matter.